A unification of the hypercontractivity and its exponential variant of the Ornstein-Uhlenbeck semigroup

Let ${\gamma }_d$ be the $d$-dimensional standard Gaussian measure and $\{Q_t{\}}_{t\ge 0}$ the Ornstein-Uhlenbeck semigroup acting on $L^1({\gamma }_d)$. We show that the hypercontractivity of ${Q_{t}}{t\ge 0}$ is equivalent to the property that \begin{align*} \left{ \int{$\mathbb{R}$^{d}}\exp \left(e^{2t}Q_{t}f\right) d\gamma_{d} \right} ^{1/e^{2t}} \le \int_{$\mathbb{R}$^{d}}e^{f},d\gamma_{d}, \end{align*} which holds for any $f\in L^1({\gamma }_d)$ with $e^f\in L^1({\gamma }_d)$ and for any $t\ge 0$. We then derive a family of inequalities that unifies this exponential variant and the original hypercontractivity, a generalization of the Gaussian logarithmic Sobolev inequality is obtained as a corollary. A unification of the reverse hypercontractivity and the exponential variant is also provided.